We develop a new algorithm for the computation of the geometrical shock dynamics model (GSD). The method relies on the fast-marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a cartesian grid. The proposed algorithm is based on a second order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure. Reference solutions are built for a smooth radial configuration and for the 2D Riemann problem. The link between the GSD model and p-systems is given. Numerical experiments demonstrate the accuracy and the ability of the scheme to handle singularities.
We study the propagation of an acoustic wave in a moving fluid in the high frequency regime. We calculate a high-frequency approximation of the solution of this problem using an Eulerian method.The model retained is a linearized Euler system around a mean fluid flow. For any regular mean flow, we derive a conservative transport equation for the geometrical optics approximation. We introduce the stretching matrix corresponding to this system, from which we deduce the geometrical spreading, key tool for computing the geometrical optics approximation.Finally, we construct and implement a numerical scheme in the Eulerian framework for the eikonal equation. This Eulerian formulation applies also for the transport equation on the stretching matrix.
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